Finite limits
Définition :
Let
be an application defined on an open interval
except in one point
with values in
We say that application
admits as limit
when
goes towards
if application
defined on
by:
is continuous in
Application
is called prolongation by continuity of application
We denote this by
Equivalent definition:
One-sided limits
We denote by
and
two real numbers,
Let
be an application defined on interval
with values in
We say that application
admits
as left-sided limit when
goes to
if:
We then write
We say that application
admits
as right-sided limit when
goes to
if:
We then write
Fondamental : Relation between one-sided limits and limit, one-sided continuity
Let
be an open interval of
an element of
and
an application from
to
Application
has a limit when
goes towards
if and only if
has a right-sided limit and a left-sided limit when
goes towards
and these limits are equal.
We then have :
Let
be a function defined on an interval
We define a function
on
by
We say that function
is continuous from the left in
if
and continuous from the right in
if
We define in a similar way the continuity from the left and from the right in one point
and function
is continuous in
if and only if it is continuous from the right and from the left in
Fondamental : Operations on limits, sum, product, quotient, composition
Let
and
be two applications defined on an open interval
except in one point
We assume :
and
then:
If we assume that
is non zero
We denote by
and
two intervals from
by
an element of
by
an element of
by
an application defined from
to
and by
an application defined from
to
We assume:
and
then:
We have the same results for one-sided limits.
